Section: Application Domains

Hybrid Systems Modelers

This is an opportunistic objective, not part of the plans stated when the team was formed. It results from a series of events: in 2008, Benoît Caillaud was part of the Synchronics large scale initiative (see section  8.1.1 ), dedicated to “embedded systems programming in 2020”. Hybrid Systems Modelers were part of the research program. Such tools are nowadays absolutely central in the development of Cyber Physical Systems (CPS), which are physical systems in closed loop with embedded control. Hybrid Systems Modelers support the modeling of physical systems (with Ordinary Differential Equations, ODE, and Differential Algebraic Equations, DAE): Matlab-Simulink and Modelica are the main players. Our vision was that these tools should deserve similar effort in theory as synchronous languages did for the programming of embedded systems. About one year after Synchronics started (focusing mostly on other topics), the PhD thesis of Simon Bliudze came to our knowledge. This thesis contained a long chapter on the use of non-standard analysis as a semantic framework for hybrid systems. The exposure relied on a recent presentation of non-standard analysis, not axiomatic in style, due to the mathematician Lindström. That attracted the attention of Albert Benveniste, so he joined the group of Synchronics working on hybrid systems. This was the beginning of a deeply novel and exciting research track.

The computer science community has devoted significant efforts to the analysis and verification of hybrid automata. The framework of hybrid automata is, however, much less flexible than what actual Hybrid Systems Modelers offer. The only ongoing effort towards modeling has been developed by Edward Lee and his team as part of the Ptolemy II project. This has led to the proposal of super-dense time semantics, in which cascades of successive instants can occur in zero time by using $R_{+}\times N$ as a time index. It turns out that the set $T=\{n\partial \mid n\in N^{*}\}$, where $\partial$ is an infinitesimal and $N^{*}$ is the set of non-standard integers is such that 1/ $T$ is dense in $R_{+}$, making it “continuous”, and 2/ every $t\in T$ has a predecessor in $T$ and a successor in $T$, making it “discrete” (le beurre et l'argent du beurre, as we say in french). Although non-effective from the operational point of view, the non-standard semantics of hybrid systems provides a framework that is very familiar to the computer scientist (who is afraid of continuous time) and at the same time efficient as a symbolic abstraction. This makes it an excellent candidate for the development of compilation schemes.